Let me try to make several statements which I believe to be true - I'm basically hoping that somebody will point out where I make an error (or errors).
1) per wikipedia, any two-level quantum mechanical system can be used as a qubit.
2) per wikipedia, an example of such a two-level QM system is given by $$ \psi(t) = M \psi(0), $$ where for all $t$ we have $\psi(t)\in \mathbb C^2$ and $M$ is the diagonal matrix $$ M := \begin{pmatrix} e^{i\omega t} & 0 \\ 0 & e^{-i\omega t} \end{pmatrix}, $$ with $\omega$ being a fixed non-zero real number.
3) when one talks about quantum computing, it is customary to use e.g. the notation $$|0\rangle =\begin{pmatrix}1 \\ 0 \end{pmatrix}$$ $$|1\rangle = \begin{pmatrix}0 \\ 1 \end{pmatrix}$$
4) It doesn't make sense to talk about this system being permanently in the state, say, $\frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$, because even if for some $t_0$ we have $$ \psi(t_0) = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle) $$ then for all $t>t_0$ which are not of the form $\frac{n\pi}{\omega}$ we have that the vectors $\psi(t_0)$ and $\psi(t)$ are linearly independent, so they correspond to different states.
Update/Clarification: This seems to me to be a problem because of the following. Suppose that we want to measure in the basis consisting of the vectors $|+\rangle:= \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$ and $|-\rangle := \frac{1}{\sqrt{2}}(|0\rangle -|1\rangle)$ . Suppose that $\omega = 2\pi$ and at $t=0$ we have $\psi(0)=|+⟩$. Then at $t=\frac{1}{4}$ we have $\psi(\frac{1}{4}) = \frac{i}{\sqrt{2}}(|0⟩−|1⟩)=i|−⟩$. It follows that at times $t=0$ and $t=\frac{1}{4}$ this "same" qubit would give different measurement outcomes with probability 1 - surely this is not how a quantum computer is supposed to work?
5) So it would seem that such a system can be used for quantum computing only under the assumption that all operations in the quantum computer are done only precisely at the times of the form $t = n\pi/\omega$ - is this right? (I know very little about engineering - but this seems to be a difficult engineering obstacle - is it not?)
6) A different idea is that, contrary to what wikipedia says, perhaps not any two-level QM system can be used as a qubit, but rather only those where the energy is the same for all states? Then it would follow that the evolution is given by a diagonal matrix with constant coefficients.
(I have some follow-up questions, but they depend on what answers I'll get).